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Planar Herding of Multiple Evaders with a Single Herder

Rishabh Kumar Singh, Debraj Chakraborty

TL;DR

This work addresses the planar herding of multiple non-cooperative evaders by a single, faster pursuer under inverse-square repulsion. It develops two pursuit laws—a spiral with radius control and a fixed-radius circular pursuit—demonstrating that all evaders converge to a limiting circular trajectory around the target, while the pursuer follows a concurrent circular path; under suitable parameter regimes these limit cycles are unique and asymptotically stable. The analysis covers both single and multiple evaders, derives existence and stability conditions, and provides numerical estimates of the regions of attraction, along with a simplified $k_1=0$ case with explicit equilibrium radii and stability. Simulation results corroborate the theoretical findings, showing controllable convergence rates and final radii via parameter choices such as $\omega$ and $R$. The results have practical implications for autonomous steering, crowd control, and search-and-harvest tasks where a single agent must effectively corral many evaders toward a target region.

Abstract

A planar herding problem is considered, where a superior pursuer herds a flock of non-cooperative, inferior evaders around a predefined target point. An inverse square law of repulsion is assumed between the pursuer and each evader. Two classes of pursuer trajectories are proposed: (i) a constant angular-velocity spiral, and (ii) a constant angular-velocity circle, both centered around the target point. For the spiraling pursuer, the radial velocity is dynamically adjusted based on a feedback law that depends on the instantaneous position of the evader, which is located at the farthest distance from the target at the start of the game. It is shown that, under suitable choices of the model parameters, all the evaders are herded into an arbitrarily small limit cycle around the target point. Meanwhile, the pursuer also converges onto a circular trajectory around the target. The conditions for the stability of these limit cycles are derived. For the circling pursuer, similar guarantees are provided along with explicit formulas for the radii of the limit cycles.

Planar Herding of Multiple Evaders with a Single Herder

TL;DR

This work addresses the planar herding of multiple non-cooperative evaders by a single, faster pursuer under inverse-square repulsion. It develops two pursuit laws—a spiral with radius control and a fixed-radius circular pursuit—demonstrating that all evaders converge to a limiting circular trajectory around the target, while the pursuer follows a concurrent circular path; under suitable parameter regimes these limit cycles are unique and asymptotically stable. The analysis covers both single and multiple evaders, derives existence and stability conditions, and provides numerical estimates of the regions of attraction, along with a simplified case with explicit equilibrium radii and stability. Simulation results corroborate the theoretical findings, showing controllable convergence rates and final radii via parameter choices such as and . The results have practical implications for autonomous steering, crowd control, and search-and-harvest tasks where a single agent must effectively corral many evaders toward a target region.

Abstract

A planar herding problem is considered, where a superior pursuer herds a flock of non-cooperative, inferior evaders around a predefined target point. An inverse square law of repulsion is assumed between the pursuer and each evader. Two classes of pursuer trajectories are proposed: (i) a constant angular-velocity spiral, and (ii) a constant angular-velocity circle, both centered around the target point. For the spiraling pursuer, the radial velocity is dynamically adjusted based on a feedback law that depends on the instantaneous position of the evader, which is located at the farthest distance from the target at the start of the game. It is shown that, under suitable choices of the model parameters, all the evaders are herded into an arbitrarily small limit cycle around the target point. Meanwhile, the pursuer also converges onto a circular trajectory around the target. The conditions for the stability of these limit cycles are derived. For the circling pursuer, similar guarantees are provided along with explicit formulas for the radii of the limit cycles.
Paper Structure (35 sections, 23 theorems, 79 equations, 11 figures)

This paper contains 35 sections, 23 theorems, 79 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Formulation
  3. Evader Kinematics
  4. Problem Formulation
  5. Coordinate Frames
  6. Evader Kinematics in $\{r,\phi\}$-frame
  7. Evader Kinematics in $\{r, \psi\}$-frame
  8. Evader Kinematics in $\{u, v\}$-frame
  9. Herding of Single Evader
  10. Equilibrium Points
  11. Existence of Equilibrium Points
  12. Uniqueness of Equilibrium Points
  13. Stability of the equilibrium point
  14. Estimation of Stable Region
  15. Translation of Results to $\{x, y\}/\{r, \phi\}$-frame
  16. ...and 20 more sections

Key Result

Lemma 2

For the system described in eq:transferred_dynamics, if $\boldsymbol{\kappa}>0$, there exist either one or three positive real roots of eq:cubic_equation, $\forall k,k_1,R,\omega>0$. Furthermore, if $2k_{1}^{2}R^{2} > 1$ and $\boldsymbol{\kappa}< \frac{\ln(2k_{1}^{2}R^{2})}{2k_{1}}$, then there exis

Figures (11)

  • Figure 1: Evader and pursuer position in Cartesian coordinate system
  • Figure 2: Estimation of $S$ in $(u,v)$-frame for $\boldsymbol{\kappa}=1, R^{\star}=1.15, \omega=2, k=1, k_1=1, R=2$. Two separate trajectories starting from $(1,0)$ and $(1,0.5216)$ converge to $(r_0^{\star}(1),\psi_0^{\star}(1))$.
  • Figure 3: Estimation of $S^{i}$ in $(u,v)$-frame for $\boldsymbol{\kappa}=1, \psi_0(0)=0, R=2, k_1=1, k=1$
  • Figure 4: Graph for $y=\cos(\psi)\sin^{2}(\psi)$
  • Figure 5: Graph of $y=r^{i*3}_s-R^{2}r^{i\star}_{s}+\frac{k}{\omega}$
  • ...and 6 more figures

Theorems & Definitions (46)

  • Lemma 2
  • proof
  • Lemma 3
  • Theorem 4
  • proof
  • Remark 5
  • Theorem 6
  • proof
  • Proposition 8
  • Lemma 9
  • ...and 36 more